{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c7eb3c96",
   "metadata": {},
   "source": [
    "# EVD SVD"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cf48c19",
   "metadata": {},
   "source": [
    "## Unitary Space 酉空间\n",
    "酉空间 是 欧氏空间 的延伸 <br>\n",
    "欧氏空间实际上是向量的带范空间<br>\n",
    "欧氏空间的向量具有 交换律、非负性以及代数线性 三个重要的性质"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ace22664",
   "metadata": {},
   "source": [
    "### 矩阵\n",
    "1. 矩阵之间的关系 - 等价 合同 相似<br>\n",
    "矩阵等价是最为广泛的概念 $$\\exists P,Q \\space s.t. B = PAQ, P,Q\\space is\\space invertable$$ <br>\n",
    "矩阵合同是一个相对强的概念，但是弱于矩阵相似 $$\\exists P \\space s.t. B=P^{T}AP,P\\space is\\space invertable$$ <br>\n",
    "矩阵相似是更强的概念 $$\\exists P \\space s.t. B=P^{-1}AP,P\\space is\\space invertable$$ <br>\n",
    "与矩阵相似相关的应用有相似对角化以及EVD SVD\n",
    "2. 合同与相似矩阵一定是等价矩阵\n",
    "3. 正交合同矩阵是相似矩阵，同理，正交相似矩阵是合同矩阵， 实际上，正交矩阵具备这样的性质: $$P^{-1}=P^{T}$$\n",
    "4. 实对称矩阵相似并且合同于对角阵，正定二次型即为实对称矩阵 实际上，实对称矩阵一定存在正交转换矩阵$P$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4d3449b",
   "metadata": {},
   "source": [
    "### 相似对角化\n",
    "1. 相似对角化实际上是特征值分解EVD\n",
    "2. 相似对角化是相似变换，具有秩不变、特征值不变、特征行列式值不变的性质\n",
    "3. 满足关系\n",
    "$$ P^{-1} \\Lambda P = B \\space,$$ wherein $\\Lambda$ is eigen value diagnal matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b284752b",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "python3.6",
   "language": "python",
   "name": "python3.6"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": true
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}
